Question

FIGURE CP22.72 shows two nearly overlapped intensity peaks of the sort you might produce with a diffraction grating (see Figure $22.8 \mathrm{b}$ ). As a practical matter, two peaks can just barely be resolved if their spacing $\Delta y$ equals the width $w$ of each peak, where $w$ is measured at half of the peak's height. Two peaks closer together than $w$ will merge into a single peak. We can use this idea to understand the resolution of a diffraction grating. a. In the small-angle approximation, the position of the $m=1$ peak of a diffraction grating falls at the same location as the $m=1$ fringe of a double slit: $y_{1}=\lambda L / d .$ Suppose two wavelengths differing by $\Delta \lambda$ pass through a grating at the same time. Find an expression for $\Delta y,$ the separation of their firstorder peaks. b. We noted that the widths of the bright fringes are proportional to $1 / N,$ where $N$ is the number of slits in the grating. Let's hypothesize that the fringe width is $w=y_{1} / N$. Show that this is true for the double-slit pattern. We'll then assume it to be true as $N$ increases. c. Use your results from parts a and b together with the idea that $\Delta y_{\min }=w$ to find an expression for $\Delta \lambda_{\operator name{mil}},$ the minimum wavelength separation (in first order) for which the diffraction fringes can barely be resolved. d. Ordinary hydrogen atoms emit red light with a wavelength of $656.45 \mathrm{nm} .$ In deuterium, which is a "heavy" isotope of hydrogen, the wavelength is $656.27 \mathrm{nm}$. What is the minimum number of slits in a diffraction grating that can barely resolve these two wavelengths in the first-order diffraction pattern? (FIGURE CAN'T COPY)

Name: Problem 72, Chapter 22: Physics for Scientist and Engineers: A Strategic Approach
Uploaded: 2020-06-28T21:34:16-08:00
Duration: 8 min 32 s
Channel: Shoukat Ali

   FIGURE CP22.72 shows two nearly overlapped intensity peaks of the sort you might produce with a diffraction grating (see Figure $22.8 \mathrm{b}$ ). As a practical matter, two peaks can just barely be resolved if their spacing $\Delta y$ equals the width $w$ of each peak, where $w$ is measured at half of the peak's height. Two peaks closer together than $w$ will merge into a single peak. We can use this idea to understand the resolution of a diffraction grating. a. In the small-angle approximation, the position of the $m=1$ peak of a diffraction grating falls at the same location as the $m=1$ fringe of a double slit: $y_{1}=\lambda L / d .$ Suppose two wavelengths differing by $\Delta \lambda$ pass through a grating at the same time. Find an expression for $\Delta y,$ the separation of their firstorder peaks.
b. We noted that the widths of the bright fringes are proportional to $1 / N,$ where $N$ is the number of slits in the grating. Let's hypothesize that the fringe width is $w=y_{1} / N$. Show that this is true for the double-slit pattern. We'll then assume it to be true as $N$ increases.
c. Use your results from parts a and b together with the idea that $\Delta y_{\min }=w$ to find an expression for $\Delta \lambda_{\operator name{mil}},$ the minimum wavelength separation (in first order) for which the diffraction fringes can barely be resolved.
d. Ordinary hydrogen atoms emit red light with a wavelength of $656.45 \mathrm{nm} .$ In deuterium, which is a "heavy" isotope of hydrogen, the wavelength is $656.27 \mathrm{nm}$. What is the minimum number of slits in a diffraction grating that can barely resolve these two wavelengths in the first-order diffraction pattern? (FIGURE CAN'T COPY)

Physics for Scientist and Engineers: A Strategic Approach

Randall Knight 2nd Edition

Chapter 22, Problem 72

Instant Answer

Solved by Expert Shoukat Ali

06/28/2020

Step 1

If two wavelengths differing by $\Delta \lambda$ pass through a grating at the same time, the position of the new peak will be $y_{1}' = (\lambda + \Delta \lambda)L / d$. The separation of their first-order peaks is $\Delta y = y_{1}' - y_{1} = \Delta \lambda L / Show more…

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FIGURE CP22.72 shows two nearly overlapped intensity peaks of the sort you might produce with a diffraction grating (see Figure $22.8 \mathrm{b}$ ). As a practical matter, two peaks can just barely be resolved if their spacing $\Delta y$ equals the width $w$ of each peak, where $w$ is measured at half of the peak's height. Two peaks closer together than $w$ will merge into a single peak. We can use this idea to understand the resolution of a diffraction grating. a. In the small-angle approximation, the position of the $m=1$ peak of a diffraction grating falls at the same location as the $m=1$ fringe of a double slit: $y_{1}=\lambda L / d .$ Suppose two wavelengths differing by $\Delta \lambda$ pass through a grating at the same time. Find an expression for $\Delta y,$ the separation of their firstorder peaks. b. We noted that the widths of the bright fringes are proportional to $1 / N,$ where $N$ is the number of slits in the grating. Let's hypothesize that the fringe width is $w=y_{1} / N$. Show that this is true for the double-slit pattern. We'll then assume it to be true as $N$ increases. c. Use your results from parts a and b together with the idea that $\Delta y_{\min }=w$ to find an expression for $\Delta \lambda_{\operator name{mil}},$ the minimum wavelength separation (in first order) for which the diffraction fringes can barely be resolved. d. Ordinary hydrogen atoms emit red light with a wavelength of $656.45 \mathrm{nm} .$ In deuterium, which is a "heavy" isotope of hydrogen, the wavelength is $656.27 \mathrm{nm}$. What is the minimum number of slits in a diffraction grating that can barely resolve these two wavelengths in the first-order diffraction pattern? (FIGURE CAN'T COPY)

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Key Concepts

Diffraction Grating

A diffraction grating is an optical device composed of many equally spaced slits that produces a diffraction pattern through interference of the light waves emanating from each slit. The pattern consists of several bright fringes or peaks, where the condition for constructive interference, typically given by d sin? = m?, results in the angular separation of these peaks. This concept is fundamental in understanding how gratings disperse light into its constituent wavelengths.

Small-Angle Approximation

The small-angle approximation simplifies the analysis of diffraction patterns by assuming that for small angles, sin? is approximately equal to tan?, which further equates to the ratio of the fringe displacement to the distance from the grating to the screen (y/L). This approximation is crucial in deriving linear relationships, such as y = (?L)/d, which relates the position of the diffraction peaks to the wavelength of the light, making complex trigonometric relations more manageable.

Interference and Fringe Formation

Interference refers to the phenomenon where waves superimpose to form a resultant wave with an amplitude that is determined by the phase difference between the individual waves. In the context of diffraction gratings, the constructive interference of light from multiple slits creates fringes or peaks in intensity. The position and spacing of these fringes depend on the wavelength of the light and the grating's properties, providing a basis for the resolution of closely spaced wavelengths.

Resolution and the Rayleigh Criterion

Resolution is the ability of an optical instrument to distinguish between two closely spaced features. The Rayleigh criterion is a standard for defining the minimum separation at which two diffraction peaks can be distinguished; typically, two peaks are just resolved when the separation between them is equal to the width of a single peak. This principle links the physical parameters of the diffraction system to its resolving power, and is central to analyzing how small differences in wavelength can be separated.

Dependence of Fringe Width on the Number of Slits

The width of the bright fringes in a diffraction pattern is inversely proportional to the number of slits in the grating. This means that as the number of slits increases, the fringes become narrower, enhancing the resolving power of the grating. Understanding this relationship is important for optimizing the design of diffraction gratings to resolve closely spaced spectral lines effectively.

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